Existence and uniqueness of the motion by curvature of regular networks

We prove existence and uniqueness of the motion by curvature of networks with triple junctions in Rd when the initial datum is of class Wp2-2/p and the unit tangent vectors to the concurring curves form angles of 120 degrees. Moreover, we investigate the regularisation effect due to the parabolic nature of the system. An application of the well-posedness is a new proof and a generalisation of the long-time behaviour result derived by Mantegazza et al. in 2004. Our study is motivated by an open question proposed in the 2016 survey from Mantegazza et al.: does there exist a unique solution of the motion by curvature of networks with initial datum being a regular network of class C 2? We give a positive answer.